Science and technology helped the United States to win the Space Race because the Space Race was all about science and technology. Simply put, the Space Race would be won by whichever country’s scientific and technological establishment was first able to solve the scientific and engineering problems that went with trying to put objects and, finally, people, into space.
In the end, it turned out that the United States had a better scientific establishment, at least if that is measured by the US’s ability to put people on the moon. The Soviet Union was able to launch an object (Sputnik) into Earth orbit first and it was able to put a person into Earth orbit before the US. However, the US then surpassed the Soviets and was able to defeat them in the race to the moon. This win also extended to the US’s supremacy in nuclear missile technology.
Science and technology helped achieve this win simply through the fact that US scientists were able to solve the problems of missile/rocket technology more quickly than the Soviets. Many people attribute this to the more open society of the United States. It is believed that American scientists were more creative because they lived in a society where individual free thinking was encouraged.